We consider a general worst-case robust convex optimization problem,
with arbitrary dependence on the uncertain parameters, which are
assumed to lie in some given set of possible values.
We describe a general method for solving such a problem, which alternates
between optimization and worst-case analysis.
With exact worst-case analysis, the method is shown to converge
to a robust optimal point. With approximate worst-case analysis, which is
the best we can do in many practical cases, the method seems to work
very well in practice, subject to the errors in our worst-case analysis.
We give variations on the basic method that can
give enhanced convergence, reduce data storage,
or improve other algorithm properties.
Numerical simulations suggest that the method finds a quite
robust solution within a few tens of steps; using warm-start techniques
in the optimization steps reduces the overall effort to a modest multiple
of solving a nominal problem, ignoring the parameter variation.
The method is illustrated with several application examples.